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String Theory on Ads3 and the Symmetric Orbifold of Liouville Theory

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String Theory on Ads3 and the Symmetric Orbifold of Liouville Theory Prepared for submission to JHEP String theory on AdS3 and the symmetric orbifold of Liouville theory Lorenz Eberhardt and Matthias R. Gaberdiel Institut f¨urTheoretische Physik, ETH Zurich, CH-8093 Z¨urich,Switzerland E-mail: [email protected], [email protected] Abstract: For string theory on AdS3 with pure NS-NS flux a complete set of DDF operators is constructed, from which one can read off the symmetry algebra of the spacetime CFT. Together with an analysis of the spacetime spectrum, this allows 3 4 us to show that the CFT dual of superstring theory on AdS3 × S × T for generic NS-NS flux is the symmetric orbifold of (N = 4 Liouville theory) × T4. For the case of minimal flux (k = 1), the Liouville factor disappears, and we just obtain the symmetric orbifold of T4, thereby giving further support to a previous claim. We also show that a similar analysis can be done for bosonic string theory on AdS3 × X. arXiv:1903.00421v1 [hep-th] 1 Mar 2019 Contents 1 Introduction1 2 Bosonic strings on AdS3 3 2.1 The sl(2; R)k WZW model and its free field realisation3 2.2 Vertex operators4 2.3 The DDF operators5 2.4 The identity operator8 2.5 The moding of the spacetime algebra8 2.6 Identifying Liouville theory on the world-sheet 11 2.7 Discrete representations 13 3 4 3 A review of superstrings on AdS3 × S × T 15 3.1 The RNS formalism 15 3.2 The hybrid formalism 17 3.3 Supergroup generators 18 4 The psu(1; 1j2)k WZW model 19 4.1 Wakimoto representation of sl(2; R)k+2 and vertex operators 19 4.2 The short representation 20 4.3 Spectral flow 21 5 The spacetime symmetry algebra 21 5.1 Spacetime operators in the RNS-formalism 22 5.2 The spacetime operators in the hybrid formalism 24 5.3 The complete spacetime algebra 26 5.4 The action of the spacetime algebra on physical states 26 6 The symmetric product orbifold 27 6.1 The T4 algebra 27 6.2 N = 4 Liouville theory 28 6.3 Identifying Liouville theory on the world-sheet 29 6.4 Spectrum generating algebra 31 6.5 The case of k = 1 31 7 Discussion 32 A Higher spin fields in spacetime 33 A.1 Internal Virasoro algebra 33 A.2 Higher spin fields 34 { i { B Various commutation relations 35 3 4 B.1 The RNS formalism of strings on AdS3 × S × T 35 B.2 The psu(1; 1j2)k WZW-model 36 B.3 The N = 4 Liouville fields 36 C Free field systems 37 C.1 bc system 37 C.2 βγ system 37 C.3 Free bosons 38 1 Introduction Holography on AdS3 backgrounds has provided us with a useful tool to test ideas about the AdS/CFT correspondence in a much more controlled setting than in the higher dimensional cases [1]. On the one hand, string theory on AdS3 backgrounds is dual to two-dimensional CFTs, which are often exactly solvable. On the other hand, string theory on AdS3 backgrounds admits an exact solution in terms of a worldsheet description employing WZW models [2{4]. This worldsheet description corresponds to a pure NS-NS flux background. It has often been asserted in the literature that the pure NS-NS background is `singular', in the sense that it features a continuum in the spectrum [5{7]. This continuum is associated to strings which can reach the boundary of AdS3 at a finite cost of energy { the so-called long strings. Hence, the dual CFT necessarily also possesses such a continuum of states, which renders the vacuum of the dual CFT non-normalisable. 3 4 It was shown in [8] that at least in the supersymmetric setting on AdS3 ×S ×T , something special happens if the NS-NS flux takes its smalles value (k = 1), in which case the strings become tensionless. In this case, the continuum vanishes completely from the spectrum due to various shortening conditions on the worldsheet. As a result the k = 1 background is dual to a bona fide CFT without a continuum. By matching the complete partition function and the fusions rules, strong evidence was given in [8] that this CFT is in fact the much-discussed symmetric orbifold of T4, see e.g. [9]. This yields an example of an AdS/CFT duality, in which both sides of the duality are exactly solvable. The purpose of this paper is twofold. First, we provide further evidence for the picture advocated in [8] by showing that not only the spectrum matches, but that we can also reproduce the algebraic structure of the dual CFT, in particular, the commutation relations of the spectrum generating fields. This goes a long way towards proving the duality in this case. The other main result is to show that a { 1 { large part of the analysis of [8] can also be done for k > 1, and that this allows us to make a convincing conjecture for the CFT dual in the more general case: for 3 4 superstring theory on AdS3 × S × T with k units of NS-NS flux, we propose that the dual CFT is the symmetric orbifold, N h i 4 Sym N = 4 Liouville with c = 6(k − 1) ⊕ T ; (1.1) in the large N limit, see also [10{14] for related work. The main idea behind elucidating the algebraic structure of the dual CFT is to construct a complete set of `DDF operators' [15] on the world-sheet. These operators commute with the physical state conditions and hence act on the space of physical states. In the context of AdS3, the construction of these DDF operators was already pioneered some time ago [16], see also [17{19] for subsequent developments. We complete and extend this analysis, and then use it to show that the symmetry algebra of the spacetime CFT is indeed the chiral algebra of the above symmetric orbifold. The worldsheet WZW model describing AdS3 is based on the affine algebra sl(2; R)k, and the full spectrum of the theory consists of a certain family of discrete and continuous representations of sl(2; R), together with their spectrally flowed im- ages. The states from the w spectrally flowed sector correspond to strings which wind asymptotically w times around the boundary of AdS3. The key observation of our analysis is to note that the moding of the DDF operators depends critically on the spectral flow sector they act on. For example, while the central charge of the spacetime Virasoro algebra `ala Brown-Henneaux [20] apparently equals c = 6kw in the w'th flowed sector [16], the modes of the spacetime Virasoro algebra are actually 1 Z 1 allowed to take values in w in this sector. This is very reminiscent of the w-cycle twisted sector of a symmetric orbifold, and we show that this interpretation is indeed correct. Furthermore, we show that a similar argument applies to all the other DDF operators (not just the spacetime Virasoro generators). We will first exemplify the construction for bosonic string theory on AdS3, where many technical complications are absent. We will show that the spectrum generating algebra of the spacetime CFT for the background AdS3 × X is that of the symmetric orbifold of the Virasoro algebra times the chiral algebra of the (arbitrary) internal CFT X. We also determine the representations of this algebra that actually appear in the spacetime spectrum, and this leads to the conclusion that the spacetime CFT is the large N limit of 6(k − 3)2 SymN Liouville with c = 1 + × X ; (1.2) k − 2 1This is the case for the continuous representations on the world-sheet; for the discrete repre- sentations the situation is more complex, see Section 2.7 and below. { 2 { i.e. the symmetric orbifold of Liouville theory times the internal CFT. Here, the continuum of Liouville theory arises precisely from the continuum of long string excitations in the bulk. While this proposal nicely encompasses all the long strings in AdS3, it does not account for the short string solutions (that arise from the discrete representations on the world-sheet). These describe non-normalizable states of the dual CFT and therefore do not explicitly appear in the CFT spectrum (in the same way as the vacuum does not appear in Liouville theory), see also [4] for a related discussion. 3 4 We also analyse in detail the supersymmetric background AdS3 × S × T , in which case the analogous conclusion to (1.2) is (1.1). We pay particular attention to the case of k = 1, for which the Liouville part becomes trivial, thus explaining the absence of the long string continuum from this viewpoint. Furthermore for k = 1, there are no discrete representations on the worldsheet [8], and hence the analysis is complete. The paper is organised as follows. In Section2 we develop the theory in the technically simpler setting of bosonic string theory on AdS3 × X. We explain how to define the DDF operators and discuss their interpretation in detail. We also comment on the role of the discrete representations (short strings) in Section 2.7. We then move on to the supersymmetric setting in Section3. It will be convenient to work at least partially in the hybrid formalism [21], as it makes spacetime supersymmetry manifest and can be used to define the k = 1 worldsheet theory. We explain carefully how the degrees of freedom of the NS-R formalism can be rewritten in terms of hybrid fields, and how this gives rise to the psu(1; 1j2)k WZW model that is discussed in detail in Section4.
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